It’s never been easier to get a ride, either on the street or through your phone. Though it’s also never been this confusing.

Until very recently, if you wanted a taxi, there was one way to get it: Go outside and stick your hand in the air. The business of the yellow cab had—from a passenger standpoint, at least—changed markedly little since 1937, when the medallion system came into use. Since then, the city has required that cabs have permits bolted to their sheet metal, and the majority of those medallions are in the hands of big taxi fleets. The business is so tightly regulated that these companies can seem state-owned, except for the fact that they make a lot of money. (One of the great art collections of the mid-20th century was assembled by taxi magnate Robert Scull.) The city has added a few medallions over the years, but not many; their essentially fixed quantity has driven prices past $1 million apiece, making their owners disinclined to risk. The fleets’ only significant change to the business model has been to shift their drivers from staff to hired-gun status, thus eliminating their benefits, some of which were later regained through unionization. A generation ago, drivers drove eight-hour shifts and earned passable middle-class wages. Today, after driving 12 hours and paying for gas and the odd parking ticket, a full-time fleet driver earns a bare-bones ­income—on the order of $35,000 per year. Neither passenger nor driver nor cabbie nor fleet owner fundamentally altered the exchange: Raise arm, pull over, drive to destination, exchange cash.

Well, as they teach in business school: If a system has chugged along, making money without challenge for decades, change will be dramatic when it does come. So it has gone with the taxicab business. The smartphone, paired with onboard electronics, opened a back route into this formerly closed world, and the yellow taxi suddenly finds itself among many other cars of unfamiliar colors.

Over the past two and half years, New Yorkers have seen the arrival of Uber, the San Francisco start-up that allows you to hail a black car, and certain yellow cabs, from your cell phone. Its junior arm, UberX, lets drivers use their own vehicles, and its passengers pay reduced rates. Lower rates, in fact, than yellow-cab riders: Uber has just dropped its X pricing by 20 percent to undercut the standard taxi rate. Only when its derided “surge pricing” comes into effect does the figuring get muddled. (But maybe not much: What are your odds of a street hail when it’s pouring?) Unsurprisingly, quite a few medallion owners have been howling about what they call unsustainable price-cutting. What they have to say about Uber’s stunts, like July ice-cream giveaways and on-­demand kitten delivery, is almost certainly unprintable.

Uber, last Friday, was joined by Lyft, a competing service that had been trying to duck New York’s Taxi & Limousine Commission regulations. Lyft has positioned itself as a more extreme UberX, letting any automobile owner who can clear a background check become a driver, picking up jobs at will. In San Francisco, driver and passenger fist-bump at the moment of pickup, a gesture intended to promote a cheery familiarity. Lyft cars are also distinguished by giant furry pink mustaches bungee-clipped to their radiator grilles. (In New York, this West Coast whimsy will likely be scaled back to a dashboard item because of concerns about grime, and also because it looks so very stupid.) A restraining order from the attorney general abruptly halted Lyft’s New York launch earlier this summer, and the company has since agreed to require drivers to have T&LC licenses, though it’s looking for a workaround that preserves its “friends driving friends” worldview. A number of other apps, like Whisk and Hailo and Gett are also carving out niches for themselves.

In the Uber-and-Lyft paradigm, the driver takes a larger cut of the car’s income than he or she would get from a yellow-cab garage, because nobody is making horrific payments on a medallion. Uber claims that the median gross income of drivers who work more than 40 hours a week is, incredibly, a bit over $90,000. The customer experience has been reimagined to be more like a livery company’s. The cars are plusher than cabs. They are unpartitioned. The Uber app displays your driver’s first name while you wait, and users seem to report more Hi, Jim! How’s traffic today? conversations than they ever had in cabs. (This may also be owed to a higher percentage of native English speakers behind the wheel. The Uber driver pool is noticeably stocked with “Who’s Who in the Cast” faces, people who might otherwise be baristas or bartenders. More women, too.) The Lyft fist bump is silly, but it does convey that this is not a relic of the old New York—where, if your cabbie greeted you with a fist, it wasn’t to say hi.